Pulsar glitches are traditionally viewed as a manifestation of vortex dynamics associated with a neutron superfluid reservoir confined to the inner crust of the star. In this Letter we show that the nondissipative entrainment coupling between the neutron superfluid and the nuclear lattice leads to a less mobile crust superfluid, effectively reducing the moment of inertia associated with the angular momentum reservoir. Combining the latest observational data for prolific glitching pulsars with theoretical results for the crust entrainment, we find that the required superfluid reservoir exceeds that available in the crust. This challenges our understanding of the glitch phenomenon, and we discuss possible resolutions to the problem.